Murisier, whose only prior World Cup podium was a giant slalom third place at Alta Badia in 2021, was third out of the gate on the Birds of Prey course and delivered a storming run of 1min 40.04sec.
Odermatt, skiing eighth, clocked 1:40.24 to finish second and Slovenian Miha Hrobat finished third in 1:40.39 in the first speed race of the season.
Swiss joy in a one-two finish was tempered by concern for 26-year-old Arnaud Boisset, who crashed heavily and was taken from the hill on a sled.
But for 32-year-old Murisier it was a long-awaited victory to cherish.
"It's amazing," he said. "I've dreamed of this for so many years. I had so many lows with injuries and surgeries. I was not even dreaming anymore of this. To achieve it now, on such a nice course, I'm super-proud."
He was also delighted to share the podium with his good friend Odermatt.
"I have to thank him. He helped me a lot when I was suffering," Murisier said.
Odermatt, who won a third straight World Cup overall title as well as the downhill, super-G and giant slalom titles last season, notched his first podium finish of the young season, having failed to finish the season-opening giant slalom at Soelden, Austria.
It was a second Beaver Creek downhill runner-up finish for Odermatt, who was second to Norway's Aleksander Aamodt Kilde in 2022.
That was the last time the men raced at Beaver Creek, with heavy snow and strong winds forcing cancellation last year.
Norway's Kilde, who completed downhill and super-G doubles at Beaver Creek in 2021 and 2022, announced in October he would miss the entire season after complications in the wake of shoulder surgery following his crash in Wengen last January.
France's Cyprien Sarrazin, who dueled with Odermatt for the downhill crown last season, finished ninth, 1.03sec behind Murisier.
American Sam Dupratt also crashed hard and was taken off on a rescue sled.
The Beaver Creek event continues with a super-G on Saturday and a giant slalom on Sunday.
